Adding further powers of the curvature scalars is not the only possible way to keep generalizing these theories. Indeed, every scalar Lagrangian of weight \(-4\) , appearing in the integral of the action, could make a candidate to a gravitational theory. A proposal whose relevance in the community has quite grown in the last few years was originally considered by Vollick [1]} and then popularized by Bañados and Ferreira [2]}, being generally known as Eddington-inspired Born-Infeld gravity (EiBI), and is given by the action:

Born-Infeld-type theories: There is quite a long tradition of invoking square-root modifications of classical actions: from the relativistic Lagrangian of point particles or the Born-Infeld modification of classical electrodynamics to cure the electron's self-energy problem [3]}, to the results of Fradkin and Tseytlin [4]} showing that actions of the Born-Infeld type arise in different scenarios related to M-theory, to finally arrive to the gravitational arena. Indeed, Born-Infeld type formulations of the gravitational field have been considered according to several frameworks and to address many astrophysical phenomena (see [5]} for a review).
\( S_{EiBI}=\frac{1}{\kappa ^2 \epsilon } \int d^4x \left[ \sqrt{-\vert g_{\mu \nu } + \varepsilon R_{\mu \nu } \vert } - \lambda \sqrt{-g} \right] + S_m(g_{\mu \nu },\psi _m)\) 
